Remark: What is, and is not, entire

The physical function ${\sum }_n{e}^{-|n+\beta |a+\mathrm{i}(n+\beta )b}$, extended by relabeling, is periodic in the flux parameter and is not the same entire branch for all complex $\beta$. The shift theorem uses $\mathcal{S}$, with a fixed splitting of positive and negative frequencies. No physical equivalence between a changed extension and a mere integer gauge transformation is asserted.